A financial institution has written f1 billion of five-year equity linked bonds with maturity guarantees. Rather than buying a five-year Put option as a hedge, it has been delta hedging the implicit option dynamically using equity index futures. Eighteen months from the outset, you have been asked to compare retrospectively the effect of the futures based strategy against what the institution would have experienced had it purchased a five-year option. You have broken down the difference between the two strategies into the main option sensitivities (Greeks) and separated the figures into three time periods: . Months 1-6 ("Market fall"), during which equity prices fell steadily. . Months 7-12 ("Volatile period"), during which equity prices were extremely volatile but finished at levels similar to those at the start of the period. . Months 13-18 ("Market recovery"), during which equity prices rose steadily. Your analysis assumes that market implied volatilities remained at 30% throughout the eighteen month period, this being the same volatility as was assumed throughout in the delta hedging calculations.1. Consider an economy with two time periods (labelled 0 and 1) in which a typical consumer has preferences given by: log(co) + Blog(c,), where ct is consumption in period f. Each consumer is endowed with ko units of capital at the beginning of period 0 and with one unit of time in each period. In each period, there is a price-taking, profit-maximizing firm that produces goods using capital and labor. These goods can be either consumed or saved in the form of capital that can be used in production in the next period. Let the firm's production function be: y = zknl-", where k is the amount of capital rented by the firm and n is the amount of labor rented by the firm in a given period. Because leisure is not valued (leisure does not appear in the utility function), each consumer supplies labor inelastically, i.c., he supplies one unit of labor in each time period. The only interesting decision that a consumer makes, then, is how much to save in period 0. Let k, be the amount of capital that a typical consumer saves in period 0. Then each consumer faces a pair of budget constraints: Co = Toko + Wo - ki ho and where r, is the rental price of capital in period / (expressed in terms of period- consumption goods) and w, is the wage rate in period / (again expressed in terms of period- consumption goods). Each consumer takes these prices as given when deciding how much to save. Because period 1 is the last period of his life, each consumer consumes all of his resources in period 1. In equilibrium, the markets for goods, labor, and capital must clear in both time periods. (a) Find an explicit expression (in terms of primitives) for the competitive equilibrium capital stock in period 1. Use your answer to determine the equilibrium rate of return on savings between periods 0 and 1. In addition, determine the equilibrium allocation of consumption across the two time periods. How do changes in z affect the equilibrium allocation? Explain. (10 points) (b) Formulate a social planning problem for this economy and show that the allocation chosen by the social planner is identical to the competitive equilibrium allocation that you determined in part (a). (5 points)Problem 2: Production economy There two firms, X and Y, and two factors of production: labor L and capital K. Firm X produces the output good X using a technology described by the following continuous, differentiable and strictly concave production function: fx (lx, kx) = Vix + Vkx where (x 2 0 is the quantity of labor employed and kx 2 0 is the quantity of capital employed by firm X. Firm Y produces the output good Y using a technology described by the following continuous and differentiable production function: fy (ly , ky) = ly + ky where ly 2 0 is the quantity of labor employed and ky 2 0 is the quantity of capital employed by firm Y. The economy has a strictly positive endowment of labor L and capital K, where L > K. The output price for good X is px > 0, and the output price for good Y is py > 0. The output prices are determined on international markets, and are parameters in this model economy. (1) Find the set of production efficient factor allocations in this econ- omy, and illustrate them in a Bowley box diagram. (2) Find the set of production efficient output allocations in this econ- omy, and illustrate the production possibility frontier in a diagram. What is the marginal rate of transformation? (3) For all possible values of the parameters (px and py) find the com- petitive equilibrium factor prices (w and r) and the competitive equilibrium allocations. What are the equilibrium profits of each of the firms?5. You are given the function g (x) =2x' -11x +10x+8 Use this information to answer each of the following questions. (a) List all the possible rational solutions to g (x). (Hint): Make sure that you use the Theorem for Bounds! [6 points] (b) Bob makes the following statement: "Based on all the knowledge we've learned in this class, I believe that that g (x) could have a total of 2 real solutions and I negative solution." Do you agree with Bob? State Yes or No and explain your answer. (Note): You need to show mathematical work in your explanation. [12 points] (c) Can we use the Intermediate Value Theorem to show that there is a root somewhere between x =0 and x = 3? If "Yes", then use the IVT to show this is true. If "No", then explain why the answer is "No". [8 points]